TA 

r 
73- 


SB    55^    5T4 

JRCHARUKJJ 


DIFEEBENT  FOBMS 


RETAINING  WALLS. 


JAM2S   S.  T\TE,  G.  E. 


f    D.   VAN 


NEW    YOTiK: 
NOSTRANIJ,   PUBLISHER, 


23    ML'KKAV    AND    H",     WARHHX 

1873. 


LIBRARY 

OF    THE 

IVERSITY  OF  CALIFORNIA 
Qa*s 


SURCHARGED 

AND 

DIFFEBENT  FOEMS 

OP 

RETAINING  ¥ALLS. 

BY 

JAMES   S.  TATE,  0.  E. 


NEW    YOBK: 
D.       IN  NOSTRAND,  PUBLISHER, 

23  MURRAY  AND  27  WARREN  STRB-KT. 

1873. 


PREFACE. 


This  little  Work  is  intended  to  supply 
what  has  no  doubt  been  often  wanted  by 
many  Engineers — a  certain  and  ready  means- 
of  correctly  and  easily  ascertaining  the  Pres- 
sures of  Embankments,  Submerged  or  other- 
wise, composed  of  different  materials ;  also 
the  Moments  of  Retaining  Walls  of  differ- 
ent forms  of  cross- section,  to  successfully 
withstand  those  pressures ;  so  that,  by  know- 
ing the  exact  value  of  each,  the  right  dimen-^ 
sions  of  the  most  suitable  form  of  wall  for 
the  purpose  required  can  be  at  once  ascer- 
t  ained. 

As  the  method  adopted  does  not  involve 
the  use  of  any  long  or  laborious  calculations, 
it  is  hoped  it  will  prove  useful  to  the  Pro- 
fession generally. 


RETAINING  WALLS. 


Retaining  walls  are  adopted  as  a  neces- 
sary expedient  in  railway  and  other  practice, 
often  under  very  peculiar  circumstances,  as 
when  there  is  not  sufficient  room  for  the 
slope  of  the  embankment ;  it  being  some- 
times perched  high  on  a  steep  mountain's 
side,  and  where  it  would  have  been  hardly 
possible  to  construct  a  railway  at  all,  except 
by  securing  it  with  a  massive  wall  occupy- 
ing comparatively  little  space. 

When  it  is  also  remembered  how  fear- 
fully terrible  any  accident  would  be  if  it ' 
was  to  occur  in  such  a  dangerous  situa- 
tion— if  by  any  erroneous   calculation   or 
mistaken  judgment  on  the  part  of  the  en- 
gineer sufficient  strength  had  not  been  given 
to  the  work,  the  wall  which  was  to  have/ 
supported  the  embankment,  suddenly  giving 
way,  falling  over  into  a  deep  ravine  or  chasm, 
a  large  portion  of  the  embankment  going 


6 


with  it,  and,  it  may  be  also,  a  passing  train 
— there  can  be  no  doubt  but  that  the  nature 
of  the  material  of  which  the  embankment 
is  to  be  made  should  be  understood,  and 
the  best*  form  and  requisite  dimensions  for 
the  wall  should  be  well  considered  and  ac- 
curately ascertained  beforehand,  so  that  it 
may  be  amply  strong  enough. 

At  the  same  time  that  the  wall  should  be 
made  perfectly  secure,  it  is  also  often  desir- 
able that  any  unnecessary  excess  of  strength 
should  not  be  given  to  it,  and  so  thereby 
avoid  increasing  its  cost  considerably,  as 
the  value  of  work  is  often  very  much  en- 
hanced when  it  has  to  be  executed  in  such 
inaccessible  situations  as  before  mentioned, 
where  all  the  materials  for  building  it  may 
have  to  be  brought  from  a  great  distance. 

The  engineer  thus  may  be  at  a  loss  to 
determine  of  what  size  a  retaining  wall 
should  be  built,  so  as  to  be  safe  against  all 
contingencies  that  can  occur,  and  yet  also 
to  be  economical. 

In  many  cases  there  have  been  failures 
which  may  have  arisen  from  not  correctly 
ascertaining  beforehand  how  the  material  of 


which  the  embankment  is  composed  will  be 
affected  by  the  alternations  of  wet  and  dry 
weather  before  it  is  thoroughly  consolidated, 
and  the  precise  angle  at  which  its  slope  will 
stand  in  either  case,  thereby  causing  a  con- 
siderable difference  in  its  pressure  against 
the  wall. 

A  retaining  wall  also,  as  in  the  case  of 
the  wing-walls  of  a  bridge,  being  built  at 
the  same  time  that  the  embankment  is  being 
filled  in  behind  it,  has  often  to  withstand 
then  a  considerable  greater  pressure  than  it 
will  have  to  do  afterwards  when  the  em- 
bankment is  settled ;  this  also  perhaps 
when  its  work  is  green,  and  not  prepared 
to  resist  the  pressure  intended  forit.  Some- 
times also  the  punning  of  the  material  be- 
hind it  has  (as  is  often  the  case)  not  been^ 
done  effectually,  and  a  heavy  rain  changes 
the  dry  Dearth  or  clay  into  a  wet  sludge, 
causing  it  to  swell  considerably. 

It  therefore  being  such  an  important 
point  in  railway  construction,  it  would  no 
doubt  be  very  desirable  if  some  simple  form 
of  calculation  were  used,  not  only  strictly 
accurate,  but  easily  adapted  to  any  circum- 


8 


stances  that  may  occur.  In  the  case  of  a 
wall  where  the  embankment  is  level  with 
its  top  the  calculation  of  the  pressure  is 
well  known,  being  very  simple,  and  is  as 
follows : 

Let  B  D  be  the  back  of  a  retaining  wall, 

D  E  the  natural  slope  of  the  embankment, 

i 

A         B  G  E 


then  if  we  bisect  the  /_  B  D  E  by  the  line 
D  G,  B  D  G  is  the  portion  of  the  embank- 
ment supported  by  the  retaining  wall. 

Now  the  weight  of  B  D  G  :  pre'ssure  of 
its  weight  against  the  wall  :  :  B  D  :.B  G  :  : 
H  :  H  tang.  L  B  D  G.  The  weight  of 


Pressure  of  weiht  of 


Moment  of  pressure  of  weight  of 

BDG  =  H'Xta"g-^BPGXjrx* 

Ja  3 

H3 

=  _  x  tang.3  L  B  D  G  X  IF, 
b 

and  the  double  of  this  moment  for  stability 


In  the  case  of  a  vertical  wall,  as  A  B  C  D, 
its  weight  =  W  H  B,  and  the  moment  of 
its  weight 

_  W  HB2 

3       »" 
then  for  equilibrium, 


-    2 —  =  —  X  tang. 2  L  B  D  G  x  W, 

'W 

n/ 
and  B  =  H  tang.  £_  B  D  G 


10 


and  for  stability, 

W  H  R2       TT3 

_J1±-  =  £L  x  tang.'  L  BDG  x  W, 
&  o 

fi~n 
V  T" 

and  JB  =  H  tang.  L  B  D  G  •• . 

V  W 

The  figures  in  the  columns  of.  Table  No. 
1,  are  calculated  from  this  last  formula, 
and  are 

/oTrf 
Htang.  Z.BDG  J™, 

BO  if  divided  by  the  square  root  of  the 
weight  of  a  cubic  foot  of  the  wall,  they  will 
give  the  thickness  of  the  wall. 

Table  No.  2  gives  double  the  moments  of 
the  pressure  of  the  weight  of  different  ma- 
terials to  form  the  embankment,  calculated 
from  the  formula 

-*  Xtang.2  ^BDGx  TF, 
o  c 

and  which,  if  made  equal  to  either  of  the 
moments  of  the  weight  of  different  forms  of 
retaining  walls  given,  the  dimensions  of 
that  form  of  retaining  wall  required  can  be 
readily  ascertained. 

Having  now  given  the  usual  formulae  and 


11 


Tables  for  easy  calculation  deduced  from 
them,  for  calculating  the  dimensions  of  a 
retaining  wall  with  an  embankment  level 
with  its  top,  what  is  next  required  is  a  con- 
venient and  ready  method  of  accurately 
calculating  the  pressure  of  a  surcharged 
embankment.  The  author  is  not  aware  if 
the  method  of  calculation  and  formulae  he 
gives  here  are  new,  but  the  Tables  for  gen- 
eral use  have,  he  thinks,  the  merit  of  sim- 
plicity. 

When  the  embankment  slopes  away  up- 
wards above  the  top  of  the  wall,  the  calcu- 
lation of  its  pressure  is  a  little  more  com- 
plex, and  no  method  of  finding  it  has  yet 
been  given  that  is  simple,  or  that  can  be 
easily  used  in  practice.  Moseley,  Hann,  and 
Rankine,  in  their  works  give  equations  very  t 
abstruse,  and  apparently  of  no  practical  ap- 
plication. Hann  also  takes  into  account 
the  pressure  of  the  slope  of  the  embankment 
resting  on  the  top  of  the  wall,  a  refinement 
of  the  calculation  practically  altogether  un- 
necessary, and  which,  by  complicating  the 
original  equation,  renders  mistakes  more 
likely  to  occur. 


12 


If  A  0  be  the  natural  slope  of  the  em- 
bankment rising  upwards  above  the  top  of 
the  wall  A  B  G  H,  B  E  a  line  parallel  to  it 
from  the  foot  of  the  wall,  B  C  bisecting  the 
L  A  B  E,  then  A  B  C  is  the  portion  of  the 
embankment  to  be  retained  by  the  wall. 
Now  when  A  B  is  vertical,  the  length  of 
the  slope  to  be  retained,  A  0,  will  be  equal 
to  the  height  of  the  wall.  If  L  E  B  F  =  the 
angle  of  the  slope  of  the  embankment  =  0, 
then 


and  if  H  =  height  of  the  wall,  then 


and  the  weight  of 


W  being  the  weight  of  a  cubic  foot  of  the 
embankment.     Pressure  of  the  weight  of 


tt  A  P 

B  A  C  = 


moment  of  pressure  of  weight  of 


13 


RAP 


^2v*  90°-0       H 

_  ^  x  tang.  __?  X  3 


WR*          /       0*" 
—  -    0^  l-4 


double  this  moment  for  stability 

"  H*         /       03-  90 

=  -^~  ^  -/  1  ~  4  X  tang,  - 


H         A 


-,'D 


7L 


G  B 

Table  No.  3  gives  the  value  of 


for  every  deg.  of  inclination  of  the  slope  of 
the  embankment  from  15  deg.  to  40  deg., 


14 


TFH3 
so  that  by  multiplying  —  —   by  this  value, 

double  the  moment  of  the  pressure  of  the 
embankment  will  be  given,  and  Table  No. 
5  gives  double  the  moments  of  different 
kinds  of  material  accordingly. 

In  the  case  of  a  vertical  wall,  the  moment 

W  H  B2 

of    its   weight   =  —  :  -  »  W   being   the 

weight  of  a  cubic  foot  of  the  wall.  Then 
for  equilibrium, 


,., 
t>  y  \y' 

and  for  stability, 


Table  No.  4  is  calculated  from  the  for- 


mula .81649  H  VcW^so  that  the  figures 
in  that  Table,  divided  by  the  square  root  of 
the  weight  of  a  cubic  foot  of  the  wall,  will 
give  the  thickness  of  the  wall  required  for 
stability. 

Table  No.  5  gives  double  the  moments  of 
the  pressure  of  the  weight  of  different  ma- 
terials to  form  a  surcharged  embankment, 


15 


with  a  retaining  wall  up  to  30  ft.  in  height, 
and  which  if  made  equal  to  either  of  the 
moments  of  the  weight  of  different  forms 
of  retaining  walls  given  afterwards,  the  di- 
mensions required  for  that  form  of  wall 
can  be  at  once  found. 

The  moment  of  a  wall  of  this  section  is 


where  B  is  the  vertical  portion  of  the  wall, 


H 


S        B 

and  S  is  the  slope.     If  S  =  1,  or  3"  to  a  ft., 
its  moment 

WH  ff^     .  H^9      H2^ 


The  moment  of  a  battering  wall  of  equal 
thickness 

WHB 
= -  J3  -f-  S  H), 


16 


where  B  =  thickness  of  wall,  and  S  H  = 
the  batter  of  the  slope  on  the  face.     If 

I  W  H  B  c^   .    HA 

S  —   r-f-  its  moment  =  —       —  I  B  +       I, 


and  if  E  F,  the  perpendicular  from  its  cen- 
tre of  gravity,  falls  on  its  inside  corner,  its 
moment  =  W  H  B2,  and  the  wall  then  will 
have  the  greatest  amount  of  resisting  power 
with  security,  and  also  with  a  minimum 
amount  of  material  in  it.  In  that  case,  if 
M  =  moment  of  earth,  W  =  weight  of  a 
cubic  foot  of  the  wall ;  for  stability, 


S  =  -y/^-ga  i  S  H  being  =  B. 

To  exemplify  this,  let  H  =  20  feet, 
S  =  -,  W  F=  sand  of  120  Ibs.  to  the  cubic 
foot  in  a  surcharged  embankment,  W  = 
brick  of  120  Ibs.  to  the  cubic  foot  in  the 


17 

wall.  Then  by  Table  No.  5,  the  double 
moment  of  that  kind  of  sand  =  160,000. 
T  hen  for  the  first  section  of  wall, 


0 


-a 


B  =  6.9.     In  this  case,  weight  of  wall 

=  120  ((20  X  6.9)  +  (1^??)J  =  22,560. 
For  second  section  of  wall, 

^^(B+2^)=  160,000, 

and  B  =  9.31,  weight  of  wall  =  9.31  X 
20  X  120  =  22344.  For  second  section 
of  wall,  and  a  perpendicular  from  its  cen- 
tre of  gravity  to  fall  on  its  inside  corner, 
120  X  20  X  ^=  160,000,  and  B  =  8.16, 
weight  of  wall  =  8.16  X  12^  X  20  =  19593 
only,  showing  a  considerable  saving  of  ma- 
terial with  this  wall. 

At  the  same  time,  though  this  wall  has 
the  greatest  amount  of  resisting  power  with 
the  smallest  amount  of  material  in  it,  yet 
perhaps  it  may  be  a  question  if  it  would  not 
be  advisable  to  make  walls  of  great  height 
thicker  from  their  base  upwards  to  one- 


18 


third  of  their  height,  which  is  the  centre  of 
pressure. 

If  we  now  consider  a  wall  of  this  form 
of  cross-section,  the  outside  slope  of  which  is 

B  0 


A  E         C        F 

S  to  1,  and  the  inside  slope  next  to  the  em- 
bankment S'  to  1,  we  find  that  its  weight  is 

WH2 


H  B  + 


(S  -  SO, 


and  the  moment  of  its  weight 


-f  B(SH  +  B)), 

or  if  we  call  it  C  E  and  0  F,  where  C  E  is 
the  difference  between  the  slopes  of  the 
front  and  back  of  the  wall,  D  E  being 
drawn  parallel  to  the  face  A  B,  and  0  F  is 


19 


the  batter  of  the  back  of  the  wall,  then  its 
weight  is 


and  the  moment  of  its  weight 


"(¥+.¥+?))•  . 

Then,  if  the  height  of  the  wall  be  20  ft., 
and  its  weight  be  120  Ibs.  per  cubic  foot,  as 

before,  its  outside  slope  —  to  1,  and  its  in- 
side slope  next  to  the  embankment  —to  1, 

then  C  F  =  2|  ft.,  E  C  =  2|  ft.,  and  its 
moment 


=  160,000,  the  double  inoment  of  the  em- 
bankment. 

From  this  equation  we  find  B  =  8.088, 
and  therefore  the  weight  of  that  wall 

=  120  X  20    s.088  +  2~\  =  22411, 


(s. 


and  which  is,  what  might  have   been  ex- 
pected from  the  'form  of  its  cross-  section, 


20 


being  between  that  of  the  first  form  of  wall 
before  mentioned,  whose  weight  was  22560, 
and  that  of  the  second  form,  whose  weight 
was  22344,  less  than  the  one  and  more  than 
the  other. 

The  form  of  cross-section  of  wall,  having 
its  front  and  back  parallel,  with  the  perpen- 
dicular from  its  centre  of  gravity  falling  on 
its  inside  corner,  having  been  proved  to  be 
the  most  economical  in  material,  it  may  be 
asked,  why  should  not  this  principle  be  car- 
ried further,  and  walls  generally  be  built 
thicker  at  the  top  than  at  the  bottom,  so  as 
to  have  their  centre  of  gravity  higher  up  ? 
This,  by  increasing  the  distance  of  a  per- 
pendicular from  it  to  the  outside  edge  of 
the  wall  at  its  foot,  would  much  increase  its 
resisting  power  to  the  overturning  force  of 
the  bank.  It  no  doubt  could  be  done,  and 
where  the  wall  is  of  great  thickness  it  may 
be  safe  to  do  so,  but  as  there  is  a  fear,  how- 
ever, of  too  much  reducing  the  thickness  of 
the  wall  at  one-third  of  its  height,  where  is 
the  centre  of  pressure,  perhaps  it  may  be 
advisable  to  make  the  form  of  equal  thick- 
ness throughout,  the  limit  of  our  endeavor 


21 


to  economize  material  with  these  forms  of 
wall. 

The  moment  of  this  form  of  wall,  with  its 
vertical  side  against  the  embankment,  is 
W  H  B2 

~~3         ' 
and  if  it  be  required  to  support  water,  whose 


double  moment  is  20.83  H3,  we  find  from 
the  equation 

^§1  =  20.83^,3  =  !^, 
V  W 

W  being  the  weight  of  a  cubic  foot  of  the 
wall. 

When  the  sloping  side  of  the  wall  is  next 
to  the  water,  the  pressure  of  the  water  on 
it  assists  the  resisting  power  of  the  wall. 

Its  moment  is 

WEB2 
6       ' 
and  the  pressure  of  the  water  on  the  slope 

S  =  62.58  x3  =31.25SH. 


22 


Thus,  when  resolved  into  the  horizontal  and 
vertical  forces,  the  former  is 

31.25  S  H  X  sin.  L  a  =  31.25  S  H  X  3  _.  31.25  H«, 

fci 

and  the  latter  is 

31.25  S  H  X  cos.  a  =  31.25  S  H  X  5  =  31.25  HB. 

to 


The  moment  of  the  former  force 

=  31.25  H2  X  ?-  =  10.416  H3, 
&     \ 

and  which  tends  to  overturn  the  wall ;  and 
the  moment  of  the  latter  force 

O  T> 

=  31 .25  H  B  X  ±f-  =  20.83  H  B2, 

and  which  tends  to  assist  the  wall.  The  total 
moment  of  the  wall  for  stability  must  there- 
fore =  2  (moment  horizontal  force  —  mo- 
ment vertical  force) 

=  2  (10.416  H3  -  20.83  H  B2)  = 
20.83  H(H2-  2B2). 


Then 


and 


WHB2 

6 


=  20.83  H(H2 -2  B2), 
11  18  H 


V  W  -|-  250  ' 

If  we  take  H  =  20  feet,  and  W  =  120  Ibs. 
per  cubic  foot,  then  in  the  first  case, 

3=14^2 =14.4-2, 


and  the  weight  of  the  wall 
120  X20X  14.42 


17304  ; 


and  in  the  second  case 
11.18X 


1/l2u  +  250 
and  the  weight  of  the  wall 


11.68, 


The  moment  of  a  wall  of  this  section  is 


24 


as  before  mentioned,  when  the  water  presses 
against  the  vertical  side,  but  if  it  is  on  the 
slope,  the  moment  is 

W  H    /"  S2>v 

^L(B(B  +  S)+-). 

If  we  have  an  embankment  of  this  form  of 


cross-section,  where  the  slopes  are  the  same 
on  both  sides,  its  moment  is 


If  the  steeper  slope  is  on  the  inside  of  the 
embankment,  its  moment  is 


S  B         S' 

If  the  steeper  slope  is  on  the  outside  of  the 
embankment,  its  moment  is 


25 


H 


S'        B  $ 

If  in  these  last  five  equations  W  =  120 
Ibs.  to  the  cubic  foot,  H  =  20  feet,  S  =  20 
feet,  S'  =  10  feet,  and  B  =  10  feet,  then 
the  moment  of  the  first  section 


"0X20 

4 


2())2 


=  920,000  ; 


of  the  second  section 


„ 

O 


of  the  third  section 

-  120x20(10  (1 

=  1,800,000;' 
of  the  fourth  section 
=  120  X  20  (10'(20  +  ! 


(-20  + 


26 

of  the  fifth  section 


In  these  equations  the  moments  of  the 
walls  are  to  be  made  equal  to  twice  the 
difference  of  the  moments  of  the  horizontal 
and  vertical  forces  of  the  water,  as  before, 
when  the  sloping  side  is  next  to  the  water. 
If  the  wall  is  to  be  built  with  a  curved 
batter  instead  of  a  slope,  to  facilitate  the 
calculation  of  its  moment  we  may  assume 
the  curve  to  be  of  a  parabolic  form,  and 
from  which,  in  the  curves  generally  used 
for  that  purpose,  it  will  not  sensibly  differ. 
The  calculations  of  the  moments  of  a  few 
forms  of  wall  with  curved  batter  are  given, 
to  show  how  they  have  been  arrived  at. 

To  find  the  moment  of  a  retaining  wall 
with  curved  batter  generally,  let  ABE  be 
of  the  parabolic  form,  then  the  area  of 


. 
o 

Now  the  centre  of  gravity  of  A  B  E  will  be 
found  sufficiently  correct  for   all  practical 


27 


purposes  if  it  is  taken  to  be  in  the  perpen- 
dicular line  G  F,  which  will  bisect  ABE. 


A      N 


£      F 


8 


Now 


HXEF- ^ 

o 


AEFG=AEFN-AGN= 
ABE       H  v 


GN  =H 


/KF 


V 


HXEF-H 


=     BE, 


2       /E~F  __  1  _    BE 
af/lBE"          6EF' 
F       9       3  BE   ,      BE2 


BE~~4       4EF^16EF2' 


28 


BE3-12BE2XEF-f  36BEXEF2-16EF3=0, 
B  E-4  E  F)  (B  E2  -  8  B  E  X  E  F  -M  E  F2)  =  0, 

BE-4EF  =  0,  BF  =  ^^• 
4 

Moment  of 


moment  of 
AECD 

moment  of 
ABCD 


If  we  take  a  triangle  of  equal  area  with 
A  G  B  E,  and  similar  to  a  triangle  ABE, 
we  shall  find  that  its  base  will  • 

=  BE  y/|  =.8165  BE, 

and  therefore  the  distance  of  a  perpendicu- 
lar from  its  centre  of  gravity  to 


O 

and  therefore,  BE-  ,2722  B  E  =  .7278  B  E 
from  B,  or  nearly  the  same  as  before.  Let 
C  E  =  6,  and  other  values  as  before,  then 


29 


160,000,  B  E  =  6.403, 
and  weight  of  wall 

—  120  Ao  X  6  +  i  20  X  6.A  =  19523. 

D  A 


To   find   the   moment   of   A  B  C  D   when 
ABE^AECD.     Then 

T)   "HI 

BE  =  3CE,  areaof  ABE  =  HX    ^===HXCE. 


Moment  of 


=  (HXCE) 


30 

.  9 


=H      CE« 


64 
Then 

120  X  20 15  B  C2  =  160,000, 
64 

B  C  =  13.62,  C  E  =  3.4,   B  E  =  10.21, 
and  weight  of  w^all 

=  120  (20  X  3.4  +  20  ^3^)  =  16344. 

When  both  the  front  and  back  of  the 
wall  are  curved  and  parallel.  When  E  F 
passes  through  the  centre  of  gravity,  to  find 
E  A.  Area  of 

E  A  B  F  =  HXEA-f-?  (B  F  -  E  A)  = 

o 


area  of 


2  H,,        .    H  r 
_BA+-g-  BF, 


E  F  C  D  =  H  X  C  F  +  -.     (E  D-C  F) 
o 


=  ) 

then,  when  E  F  bisects  A  B  C  D, 


31 


For  stability, 


SBC    SAD 


OF  --- 

4  4 

=  ^?   +  2(ED-EA). 

r>        E  xi 


and  as 


=  8EA,    EA=-AD. 


To  find  E  A  when  the  perpendicular 
which  bisects  A  B  C  D  passing  through  its 
centre  of  gravity  falls  on  its  inside  corner. 
Area  of 

=  ?-H(AD-EA). 


Area  of 


32 


(A  D  -  E  A)  =  H  X  E  A  -f-  5.  (B  C  -  E  A), 

O 


C  .5 

To  find  the  moment  of  A  B  C  D  when 
the  curves  of  the  front  and  back  of  the  wall 
are  of  different  radii.  Area  of 


EF 


2H, 


C  D  =  H  X  C  F  +  -0-  (D  E  -  C  F) 
o 


33 


area  of 


area  of 


=  HXEA-f-(BF-EA) 
o 

2H  H 

=  __  E  A  +  -  B  F, 


~ 
o 


-  B  F  =        (B  C  +  2  D  A)  ; 


£      A 


moment  of  A  B  C  D  for  stability 


As 


34 

adding  2  E  A  to  both  sides, 

C  F  +  2  DA  =  4  EA  +  B  F, 
2D  A+CF-BF 
~T~ 

generally,  and  for  stability, 
BC      3BC 

EA-DA  4-  -i-      -±-   -^A  _*^ 

T7        44  28' 

If  D  A  is  to  be  —  B  C,  then  the  moment 


will  be 


HXBC  /  3         \  5BC2 

3  +         °  ~~"* 


Then,  with  values  as  before, 

120  X  20  5-?r^  =  160,000,  B  C  =  10.32, 

o 


=  -      10.32  =  7.74, 


weight  of  wall 


90 

=  120  X  ^  (10.32  -f  2  X  7.74)  =  20640. 
o 

If  D  A  is  to  be  4-  B  C,  then  the  moment 

o 


HX.BC  (B  C  +  2  0  A) 


will  be 


35 


Then,  with  values  as  before, 

120  X  20  —   -  =  160,000,  B  C  =  10.69, 
12 

D  A  =  4-  10.69=  7.13, 
o 

weight  of  wall 

=  120  X  ^  (10.69  +  2  X  7.13)  =  19960. 
rf 

If  D  A  is  to  be  1  B  C,  then  the  moment 
2 


will  be 

?2iBC(BO  +  BC)  =  H]^. 

Then,  with  values  as  before, 

120  X  20  §~-  =  160,000,  B  C  =  11.54, 
D  A  =  i  11.54  =  5.77, 

a 

weight  of  wall 

20 

=  120  X  ~  (11.54  -f-  2  X  5.77)  =  18464. 
o 

If  D  A  is  to  be  —  B  C,  then  the  moment 
TT  x  ~R  P 

will  be 


36 


Then,  with  values  as  before, 

120  X  20  —  —  =  160,000,  B  C  ==  13.3, 


=  ~  13.3  =  3.3, 


weight  of  wall 

20 
=  120  X  ~  (13.3  +  2  X  3.3)  =  16000. 

o 

If  a  wall  of  this  section  is  required,  its 
moment  is 


and  if  it  supports  water  level  with  the  top, 


T)   /~«2 

120  X  20  - ~  =  166,666,  B  C  =  16.6  ; 
•and  weight  of  wall 


37 


=  120  X  20  X 


=  13333. 


Now  as  17304  was  required  for  the  trian- 
gular form  of  wall  with,  the  same  values, 
there  is  shown  to  be  a  great  saving  of  ma- 
terial with  the  form  of  wall  with  curved 
batter. 

As  the  form  of  wall  with  a  curved  batter 
of  the  semi-  cubical  parabolic  section,  has 
been  proved  by  several  writers  to  be  every- 
where of  equal  strength,  the  calculations 
for  finding  the  dimensions  of  retaining 

AN  K 


£     £  8 

walls  with  a  batter  of  that  curve  are  also 
given,  as  they  may  be  found  useful  in  some 
cases.  Let  A  G  B  in  this  figure  be  a  curve 


38 


of  that  form,  with  G  F  passing  through  its 
centre  of  gravity.     Then 

2  =  B  E2  :  B  K3  =  H3  :  :  A  H2  =  E  F2  :  G  N3, 


/E  F2 


area  of        A  G  N  =  3  A^ x  G  H 

5  ' 

~~  If 
~  ~5~  "~  ~&\ 

o  ^  /V    T?2  TJ 

HXEF-   ^EFXH./  =4«  =  ? 

O  r      X>  Jli  O 

3  3/El^"        B  E 

5  v    JD  lj 


5  *"   ^   V  BE2 

EFi/: 


BE2  5    ' 

BE 

5EF      BE 


BE2 


3 


BE 


/E  F2       5         BE 

V  Bl^  =  3~  ""  ?E^ ¥" 


A    B^3   A    B^ 
V^EF)     (5-Wp) 


_ 
BE2 


27 


_  _27 

EF*' 


39 


BE*/.     BEV«       27      BE 
=  27' 


B  E  =  3.759  E  F  =  3.759  (B  E  -  B  F), 
3.759  B  F  =  3.759  B  E  -  B  E, 

2.  759  BE 
3.759      = 

Moment  of 


A  B  E  =          B  E  X  .734  B  E  =  .2936  B  E*  X  H  ; 
5 

moment  of 


moment  of 

ABCD  =H  (^~-f  CEXBE  +  .2930BEA 

=  -5  ((0  E  +  B  E)2  -  .4f28  B  E2). 
a 

Let  C  E  =  6,  and  other  values  as  before, 
then 

20 
120  —  ((6  +  B  E)2  -  .4128  B  E2)  =  160,000, 

m 

B  E  =  6.22,  and  weight  of  wall 

=  120  (20  X  6  +  -|  20  X  6.22")  =  20369. 
To  find  the  moment  of  A  B  C  D  when 


40 


ABE  =  AECD.    Then  B  E  =    -  C  E, 


area  of  A  B  E  =  H  X  -    B  E  =  H  X  C  E. 


(.734  B  E)  =  (H  X  C  E) 


|  C  E  +  5^)  +  (H  X  C  E)  (.  734  X  |  C  E) 


=  (HXCE)3CE  +  (HXCE)  1.835CE 
=  4.835  HXCE2.    Then  120X20  x  4.835  CE2 

=  160,000,  CE  =  3.71,  BE=|-  3.71  =  9.28, 

SB 

and  weight  of  wall 

=  120  X  20  (3.71  +  -|  9.28^  =  17808. 


41 


When  both  the  front  and  back  of  the 
wall  are  curved  and  parallel.  When  E  F 
passes  through  the  centre  of  gravity,  to  find 
E  A.  Area  of 


H  . 


-•/<•»+.!?«* 


E      A 


area  of 


OF  B 

then  when  E  F  bisects  A  B  0  D, 

^^•T?Ai^-^T5T?  ^-^-r«T?l^^-l?Tk 

—  E  A  +  -g-  B  F  =  -6-  0  F  -I-  -y-  E  D, 


42 


for  stability, 


SO  3  A  L>  _  A  JLJ  +  3  j 

AD  =  3ED-3EA,  and  as 
subtracting, 

:  6  E  A,  E  A  = 


=  3ED  +  E  A, 


To  find  E  A  when  the  perpendicular 
which  bisects  A  B  0  D  passing  through  its 
centre  of  gravity,  falls  on  its  inside  corner. 
Area  of 


area  of 


2H 
5 


43 


~  (A  D  -  E  A)  ==  H  x  E  A  +  ?_?  (B  C  -  E  A) 
o  o 

3H  2H 


To  find  the  moment  of  ABCD  when 
the  curves  of  the  front  and  back  of  the  wall 
are  of  different  radii.  Area  of 

EFCD^HXCF  + 


area  of 


area  of 


5 
3H^        ,    2H, 


=  —  OF+  ^F 

£>  o 


—  EA+^BF  =  ^(2BC 
o  o  o 

moment  of  A  B  0  D  for  stability 


44 
=  ^g^(2BC  +  3AD). 


D 


E     A 


C  B 

adding  3  E  A  to  both  sides, 


3DA+2CF-2BF 
EA=-  —  g- 

generally,  and  for  stability, 

BC      3BC 
3PA  +  lr       ^-        DA        BC 

~6~  ~2~        ~6~' 

q 

If  D  A  is  to  be  —  B  0,  then  the  moment 
3  H     B  C 


(2  B  C  +  3  A  D) 


will  be 


45 


Then,  with  values  as  before, 

120  X  20  51  ^°2  =  160,000,  B  C  =  10.23, 

oU 

D  A  =  f-  10.23  =  7.67, 

4 

weight  of  wall 

=  120  X  ^  (2  X  10.23  +  3  X  7.67)  =  208CO. 
5 

2 
If  DA  is  to  be  -    B  C,  then  the  moment 


will  be 


Then,  with  values  as  before, 

q   T>    p2 

120  X  20  =  160,000, 

o 

B  C  =  10.54,  D  A  =  1-10.54  =  7.03, 
o 


weight  of  wall 

90 

=  120  X  ^  (2  X 
5 

If  D  A  is  to  be  -^  B  0,  then  the  moment 


20 

=  120  X  -  (2  X  10.54  +  3  X  7.03)  =  20238. 
o 


46 

3HX  BC 


will  be 


20 


(2  B  C  -f  3  A  D) 


Then,  with  values  as  before, 

120  X  20  21  ^°2  =  160,000, 
40 

B  C  =  11.27,  D  A  =  i-  11.27  =  5.63, 
SB 

weight  of  wall 

90 

=  120  X  =p  (2  X  11.27  +  3  X  5.63)  =  18930. 
o 

If  A  D  is  to  be  -5-  B  C,  then  the  moment 


3 
3HXB  C 


20 


(2  B  0  -f  3  A  D) 


will  be 

3H  x  B  C 


. 

C2  B  o  -f 


~ao 
Then,  with  values  as  before, 

n  T>  r]2 

120x20     "         =160,000, 


B  C  =  12.17,  D  A  =       12.17  =  4.06, 
3 


weight  of  wall 


90 

120  X  TT   (a  X  12-17  +  3  X  4-06>  =  17526' 
D 


47 


If  D  A  is  to  be  -j-  B  C,  then  the  moment 

3  H  x  B  C  0  ^  ~  _ 

— — (2  B  C  -f  3  A  D) 


(2BC  +  |Bc),H2Si51. 


will  be 
3HxBC 

Then,  with  values  as  before, 

OO    T>    p2 

120  X  20  =  160,000, 

do 


B  C  =  12.71,  D  A  =  -1 12.71  =  3.18, 
weight  of  wall 

20 
;=  120  X  ^  (2  X  12.71  +  3  X  3.18)  =  16780. 

If  a  wall  of  this  section  is  required,  its 


48 

moment  is  .2936  B  C2  X  H>  an(i  if>  &  sup- 
ports water  level  with  the  top, 

120  X  20  x  .2936  B  C2  =  166,666,  B  C  =  15.38, 
and  weight  of  wall 

=  120  +  20X2X^38=  14764 
5 

Having  now  given  methods  for  finding 
the  correct  dimensions  of  the  different  forms 
of  wall  that  are  generally  used  in  practice, 
the  author  does  not  wish  to  express  any 
opinion  on  the  merits  of  any  particular 
form  of  wall, « leaving  it  to  the  superior 
judgment  of  more  experienced  engineers  to 
determine  the  section  of  wall  they  may  con- 
sider most  suitable  in  each  case. 


50 


TABLE  1. — Thickness  of  Vertical  Retaining  Walls, 
to  sustain  the  Pressure  of  Earth,  Sand,  etc.,  level 
with  its  top.  The  Moment  of  the  Wall  is  equal 
to  twice  that  of  the  Earth,  etc.,  to  insure  perma- 
nent stability. 


Sand. 

Shingle. 

Dry  earth. 

*c 

£  =  30°. 

A  =  400. 

A  =  43°. 

11 

<u 
W 

94  Ibs. 

120  Ibs. 

119  Ibs. 

106  Ibs. 

94  ibs. 

6 

27.42 

30.98 

24.92 

23.62 

20.65 

7 

31.99 

36.15 

29.07 

27.44 

24.09 

8 

36.56 

41.31 

33.23 

31.36 

27.53 

9 

41.13 

46.47 

37.38 

35.28 

30.98 

10 

45.70 

51.64 

41.53 

39.20 

34.42 

11 

50.27 

56.80 

45.69 

43.12 

37.  SB 

12 

54.84 

61.97 

49.84 

47.04 

41.30 

13 

59.42 

67.13 

53.99 

50.96 

44.74 

14 

63.99 

72.29 

58.15 

64.88 

48.19 

15 

68.56 

77.46 

62.30 

58.80 

51.63 

16 

73.13 

82.62 

66.45 

62.72 

55.07 

17 

77.70 

87.79 

70.61 

66.64 

58.51 

18 

82.27 

92.95 

74.76 

70.56 

61.95 

19 

86.84 

98.11 

78.91 

74.48 

65.40 

20 

91.41 

103.28 

'  83.07 

78.40 

68.84 

21 

95.98 

108.44 

87.22 

82.32 

72.28 

22 

100.55 

113.61 

91.38 

86.24 

75.72 

23 

105  .  12 

118.77 

95.53 

90.16 

79.17 

24 

109.69 

123.94 

99.68 

94.08 

82.61 

25 

114.26 

129.10 

103.84 

98.00 

86.05 

26 

118.83 

134.26 

107.99 

101.92 

89.49 

27 

123.40 

139.43 

112.14 

105.84 

92.93 

28 

127.97 

144.59 

116.30 

109.76 

96.38 

29 

132.54 

149.76 

120.45 

113.68 

99.82 

30 

137.11 

154.92 

124.60 

117.60 

103.26 

51 


TABLE  1. — Continued. 


Do.,  moist  or 
natural. 

Do.  ,  dense  and 
compact. 

Clay, 
o 

Clay. 

A  =  54°. 

L  =  55o. 

L- 

106  Ibs. 

125  Ibs. 

125  Ibs. 

125  Ibs. 

16.39 

17.27 

41.27 

22.69 

19.12 

20.15 

48.15 

26.47 

21.85 

23.03 

55.03 

30.25 

24.58 

25.90 

61.91 

34.03 

27.31 

28.78 

68.79 

37.81 

30.04 

31.66 

75.67 

41.59 

32.78 

34.54 

82.55 

45.38 

35.51 

37.42 

89.43 

49.16 

38.24 

40.29 

96  31 

52.94 

40.97 

43.17 

103.18 

56.72 

43.70 

46.05 

110.06 

60.50 

46.43 

48.93 

116.94 

64.28 

49.16 

51.81 

123.82 

68.06 

51.89 

54.69 

130.70 

71.84 

54.63 

57.56 

137.58 

75.62 

57.36 

60.44 

144.46 

79.40 

60.09 

63.32 

151.34 

83.18 

62.82 

66.19 

158.22 

86.96 

65.56 

-69.07 

165.10 

90.74 

68.29 

71.95 

171.97 

94.52 

71.02 

74.83 

178.85 

98.31 

73.75 

77.71 

185.73 

102.09 

76.48 

80.58 

192.61 

105.87 

79.21 

33.46 

199.49 

109.65 

81.94 

86.34 

206.37 

113.43 

52 


TABLE  2. — Double  Moments  of  the  Pressure  of  the 
Weight  of  Embankments  of  Earth,  Sand,  etc., 
level  with  the  top  of  Wall. 


Shingle. 

Dry  earth. 

Sand. 

cTFH* 

=10.4  H« 

13.3H3 

8.62522  H:* 

7.  6829  H  3 

5.  92394  H^ 

3 

6 

2256 

2880 

1863 

1659 

1280 

7 

3582 

4573 

2958 

2635 

2032 

8 

5347 

6827 

4416 

3934 

3033 

9 

7614 

9720 

6287 

5601 

4318 

10 

10444 

13333 

8625 

7683 

5924 

11 

13901 

17747 

11480 

10226 

7885 

12 

18048 

23040 

14904 

13276 

10236 

13 

22946 

29293 

18949 

16879 

13015 

14 

28659 

36587 

23667 

21081 

16255 

15 

35250 

45000 

29110 

25929 

19993 

16 

42780 

54613 

35329 

31468 

24264 

17 

51313 

65507 

42376 

37745 

29104 

18 

60912 

77766 

50302 

44805 

34548 

19 

71638 

91453 

59160 

52696 

40632 

20 

83555 

106666 

69002 

61461 

47391 

21 

96726 

123480 

79878 

71149 

54862 

22 

111212 

141973 

91841 

81805 

63078 

23 

127077 

162227 

104943, 

93475 

72077. 

24 

144384 

184320 

119235 

106206 

81892 

25 

163194 

208333 

134769 

121042 

92561 

26 

183571 

234346 

151597 

135035 

104119 

27 

205578 

262440 

169770 

151222 

116601 

28 

229276 

292693 

189341 

168655 

130042 

29 

254729 

325186 

210360 

187378 

144479 

30 

282000 

360000 

232881 

207438 

159946 

53 


TABLE  2. — Continued. 


Do.,  moist 
or  natuial. 

Do.  ,  dense 
and 
compact. 

Clay. 

Water. 

3.  73024  H  3 

f 
4.  14222  H  3 

23.66012  H  3 

7.14887  H  3 

20.83  H  3 

806 

895 

5110 

1544 

4500 

1279 

1421 

8115 

2452 

7146 

1910 

2121 

12114 

3660 

10666 

2719 

3020 

17243 

5211 

15187 

3730 

4142 

23660 

7149 

20833 

4965 

5513 

31492 

9515 

27729 

6446 

7158 

40885 

12353 

36000 

8195 

9100 

51981 

15706 

45771 

10236 

11366 

64923 

19616 

57166 

12590 

13980 

79853 

24127 

70312 

15279 

16966 

96912 

29282 

85333 

18327 

20251 

116242 

35122 

102354 

21755 

24157 

137986 

41692 

121500 

25586 

28411 

162285 

49034 

142896 

29842 

33138 

189281 

57191 

166666 

34546 

38361 

219116 

66206 

192937 

39720 

44106 

251933 

76121 

221833 

45386 

50398 

287873 

86980 

253479 

51567 

57262 

327077 

98826 

288000 

58285 

64722 

369689 

111701 

325521 

65563 

72804 

415850 

125650 

366166 

73422 

81531 

465702 

140711 

410062 

81886 

90930 

519387 

156932 

457333 

90977 

101025 

577046 

174354 

508104 

100716 

111840 

638823 

193019 

562499 

54 


TABLE  3. — For  Surcliarged  Embankments. 


£  of  slope  =  0. 

U,c. 

»•;, 

Tine1  nf 

2 

aue»*  u     't 

4  tol 

=  14°  12' 
15   0 
16  0 

37° 
37 
37 

54/ 
30 
0 

.77847 
.  76732 
.75355 

17   0 

36 

30 

.  73996 

18   0 

36 

0 

.72654 

3  tol 

=  18  25 

35 

47^ 

.72100 

19   0 

35 

30 

.71329 

20  0 

35 

0 

.  70020 

21   0 

34 

30 

.68728 

22   0 

34 

0 

.67450 

23  0 

33 

30 

.66188 

24  0 

33 

0 

.64940 

25   0 

32 

30 

.63707 

26  0 

32 

0 

.62486 

2  tol 

=  26  35 

31 

42^ 

.61781 

27  0 

31 

30 

.61280 

28   0 

31 

0 

.60086 

29   0 

30 

30 

.58904 

IK  tol 

=  29  44 

30 

8 

.58045 

30   0 

30 

0 

.57735 

31   0 

29 

30 

.56577 

32   0 

29 

0 

.55430 

33  0 

28 

30 

.54295 

\y  to  i 

=  33  42 

28 

9 

.53507 

34   0 

28 

0 

.53170 

35   0 

27 

30 

.52056 

36   0 

27 

0 

.50952 

37   0 

26 

30 

.49858 

38   0 

26 

0 

.48773 

l^tol 

=  38  40 

25 

40 

.48055 

39   0 

25 

30 

.47697 

40   0 

25 

0 

.46630 

55 


TABLE  3.—  Continued. 


o   s  n.  (90°  +  0) 



Taog.  of  9°°  -  * 

u  —  '•  — 

o2 

.    2 

8i,  (^) 

*     -  "4 

(»Vl-T) 

1.5782 
1.5867 

0.96944 
0.96592 

c 
0.75469 
0.74118 

1.5972 

0.96126 

0.72436 

1.6077 

0.95630 

0.70762 

1.6180 

0.95105 

0.69098 

1  .  6223 

0.94878 

0.68407 

1.6282 

0.94551 

0.67443 

1.6383 

0.93969 

0.65798 

1.6483 

0.93358 

0.64163 

1.6581 

0.92718 

0.62539 

1.6678 

0.92050 

0.60926 

1.6773 

0.91354 

0.59326 

1.6868 

0.90630 

0.57738 

1.6961 

0.89879 

0.56162 

1.7014 

0.89428 

0.55250 

1.7053 

0.89100 

0.54601 

1.7143 

0.88294 

0.53052 

1     1.7232 

0.87461 

0.51519 

1  .  7297 

0.86834 

0.50403 

1.7320 

0.8C602 

0.50000 

.7407 

0.85716 

0.48496 

.7492 

0.84804 

0.47008 

.  7576 

0.83867 

0.45536 

.7634 

0.83195 

0.44515 

.7659 

0.82903 

0.44080 

.7740 

0.81915 

0.42642 

1.7820 

0.80901 

0.41221 

1.7899 

0.79863 

0.39818 

1.7976 

0.78801 

0.38433 

1.8026 

0.78079 

0.37521 

1.80517 

0.77714 

0.37067 

1.81261 

0.76604 

0.35721 

56 


TABLE  4. — Thickness  of  Vertical  Retaining  Walls 
to  sustain  the  Pressure  of  a  Surcharged  Em- 
bankment of  Earth,  Sand,  etc.  The  moment  of 
the  Wall  is  equal  to  twice  that  of  the  Earth,  etc., 
to  insure  permanent  stability. 


§ 

g 

0 

1 

Sand.            « 

Shingle.          g 

Dry  earth,  co 

p 

A  =  300.         o 

^=40°.         o 

A  ^  43°.     co 

o 

0 

li 

|| 

•E 

0 

o 

"5 
B 

94  Ibs. 

120  Ibe. 

119  Ibs. 

106  Ibs. 

94  Ibs. 

6 

33.58 

37.94 

31.94 

30.14 

26.78 

7 

39.18 

44.27 

37.26 

35.17 

31.24 

8 

44  78 

50.59 

42.58 

40.19 

35.71 

9 

50.37 

56.92 

47  91 

45  21 

40.17 

10 

55.97 

63.24 

53.23 

50  24 

44.64 

11 

61  57 

69.57 

58.55 

55.26 

49.10 

12 

67.17 

75.89 

63.88 

60.29 

53.56 

13 

72.76 

82.21 

.69.20 

65.31 

58.03 

14 

78.36 

88.54 

74.52 

70.33 

62.49 

15 

83.96 

94.86 

79.85 

75.36 

66.96 

16 

89.56 

101.19 

85.17 

80.38 

71.42 

17 

95.16 

107.51 

90.49 

85.41 

75.89 

18 

100.75 

113.84 

95.82 

90.43 

80  35 

19 

106.35 

120.16 

101.14 

95.46 

84.81 

20 

111.95 

126.49 

106.46 

100.48 

89.28 

21 

117.55 

132.81 

111  .  79 

105  50 

93.74 

22 

123.14 

139.14 

117.11 

110.53 

98.21 

23 

128.74 

145.46 

122.43 

115.55 

102.67 

24 

134  34 

151.78 

127.76 

120.58 

107.13 

25 

139.94 

158.11 

133.08 

125.60 

111.60 

26 

145.54 

164.44 

138.41 

130.63 

116.07 

27 

151.13 

170.76 

143.73 

135.65 

120.53 

28 

156.73 

177.09 

149.05 

140.68 

124.99 

29 

162.33 

183.41 

154  38 

145.70 

129.46 

30 

167.93 

189.74 

159.70 

150.73 

133.92 

57 


TABLE  4. — Continued. 


CO 

Do->         g 

Do.,  dense  S? 

moist  or    § 

and        g 

Clay. 

Clay. 

natural.     ^ 

compact.    S 

L  =  16°. 

L  =  ^Q. 

Z.  -  54°.      || 

L  -55°.     ,| 

0 

w 

106  Ibs.  ^ 

125  Ibs. 

125  Ibs. 

125  Ibs. 

22.04 

23.29 

46.61 

29.64 

25.71 

27.17 

54.38 

34.58 

29.39 

31.05 

62.15 

39.52 

33.06 

34.93 

69.92 

44.46 

36.73 

38.82 

77.69 

49.40 

40.41 

42.70 

85.46 

54.34 

44.08 

46.58 

93.23 

59.28 

47.75 

50.46 

101.00 

64.22 

51.43 

54.34 

108.77 

69.16 

55.10 

58.23 

116.54 

74.10 

58.78 

62.11 

124.31 

79.04 

62.45 

65.99 

132.08 

83.98 

66.12 

69.87 

139.85 

88.92 

69.80 

73.75 

147.62 

93.86 

73  47 

77.64 

155.39 

98.80 

77.14 

81.52 

163.16 

103.74 

80.82 

85.40 

170.93 

108.68 

84.49 

89.28 

178.70 

113.62 

88.16 

93.16 

186.47 

118.56 

91.84 

97.05 

194.24 

123.50 

95.52 

100.93 

202.01 

128.45 

99.19 

104.81 

209  .  78 

133.39 

102  86 

108.70 

217.55 

138.33 

106.54 

112.58 

225.32 

143.27 

110.21 

116.46 

233.10 

148.21 

58 


TABLE  5. — Double  Moments  of  the  Pressure  of  tfie 
Weight  of  Surcharged  Embankments  of  Earth, 
Sand,  etc. 


Sand. 

Shingle. 

cTFHs 

20  H3 

14.  16945  H  3 

12.  62153  H  3 

3 

6 

3384 

4320 

3061 

2726 

7 

5373 

6860 

4860 

4329 

8 

8021 

10240 

7255 

6462 

9 

11421 

14580 

10330 

9201 

10 

15666 

20000 

14169 

12621 

11 

20852 

26620 

18860 

16799 

12 

27072 

34560 

24485 

21810 

13 

34419 

43940 

31130 

27729 

14 

42989 

54880 

38881 

34633 

15 

52875 

67500 

47822 

42598 

16 

64170 

81920 

58038 

51698 

17 

76970 

98260 

69614 

62010 

18 

91368 

116640 

82636 

73609 

19 

107457 

137180 

97188 

86571 

20 

125333 

160000 

113355 

100972 

21 

145089 

185220 

131223 

116888 

22 

166818 

212960 

150876 

134394' 

23 

190616 

243340 

172400 

153566 

24 

216576 

276480 

195878 

174480 

25 

244791 

312500 

221397 

197211 

26 

275357 

351520 

249042 

221836 

27 

308367 

393660 

278897 

248429 

28 

343914 

439040 

31  1048 

277068 

29 

382094 

487780 

345579 

307826 

30 

423000 

540000 

382575 

340781 

59 


Do.,  moist 

Do.,  dense 

Dry  earth. 

or 

and 

Clay. 

natural. 

compact. 

9.96405  H3 

6.  748066  H  3 

7  53526  H3 

30.183  H  3 

12.  20375  H* 

2152 

1458 

1627 

6520 

2636 

3418 

2315 

2585 

10353 

4186 

5102 

3455 

3858 

15454 

6248 

7264 

4919 

5493 

22004 

8896 

9964 

6748 

7535 

30183 

12204 

13262 

8982 

10029 

40174 

16243 

17218 

11661 

13021 

52157 

21088 

21891 

14825 

16555 

66313 

26812 

27341 

18517 

23677 

82823 

33487 

33628 

22775 

25432 

101869 

41188 

40813 

27640 

30864 

123631 

49986 

48953 

33153 

37021 

148291 

59957 

58110 

39355 

43946 

176029 

71172 

68343 

46285 

51684 

207027 

83712 

79712 

53985 

60282 

241466 

97630 

92277 

62494 

69784 

279528 

113019 

106097 

72453 

80235 

321392 

1:9945  * 

121232 

82104 

91681 

367241 

148483 

137743 

93285 

104167 

417254 

168704 

155688 

105438 

117738 

471615 

190684 

175128 

118604 

132439 

530502 

214493 

196122 

132822 

148316 

594098 

240206 

218731 

148133 

165414 

662584 

267897 

243013 

164578 

183777 

736141 

297637 

269029 

182198 

203452 

814949 

329501 

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THIS  BOOK  IS  DUE  ON  THE  LAST  DATE 
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CHEMICAL  PROBLEMS.     By  Prof   Foye 
70. -EXPLOSIVE  MATERIALS.    By  M.  P  fi  Berth 

HopMu 


in    u  i 

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Prof.  A.  S.  Hardy,  John  B.  McMaster  and  H.  F. 
Walling. 

x  73.— SYMBOLIC  ALGEBRA  ;  or  The  Algebra  of  Algebraic 
Numbers.  By  Prof .  W.  Cain. 

).  74.— TESTING  MACHINES;  their  History,  Construction 
and  Use.  By  Arthur  V.  Abbott. 

).  75.— RECENT  PROGRESS  IN  DYNAMO-ELECTRIC  MA- 
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tric Machinery.  By  Prof.  Silvanus  P.  Thompson. 

).  76. -MODERN  REPRODUCTIVE  GRAPHIC  PRO- 
CESSES. By  Lt.  Jas.  S.  Pettit,  U.S.A. 

).  77. -STADIA  SURVEYING.  The  Theory  of  Stadia 
Measurements.  By  Arthur  Winslow. 

>.  78.— THE  STEAM  ENGINE  INDICATOR,  and  its  Use. 
By  W.  B.  Le  Van. 

».  79.-THE  FIGURE  OF  THE  EARTH.  By  Frank  C. 
Roberts,  C.  E. 

».  80.— HEALTHY   FOUNDATIONS    FOR   HOUSES.     By'' 
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.  81.— WATER  METERS:  Comparative  Tests  of  Accu*- 

Delivery,  etc.    Distinctive  Features  of  tin    ' 

ington,  Kennedy,  Siemens  and  Hesse  ¥ 

Ross  E.  Browne. 
.  82 —THE  PRESERVATION  OF   TIMBER, 

of  Antiseptics.    By  Samuel  Bagster ' 
.  83.-MECHANICAL    INTEGRATORS,    k 

S.  H.  Shaw,  C.  E. 
.  84.-FLOW  OF  WATER  IN  OPEN  CHAr 

CONDUITS,    SEWERS,   &c. ;    - 

P.  J.  Flynn,  C.  E. 

.  85.-THE  LUMINIFEROUS  JETHF" 

son  Wood. 
.  86.— HANDBOOK    OF    MINER 

and  Description  of  Mr 

States.    By  Prof.  *  r 
87.— TREATISE    ON    T 

STRUCTION 

ARCHES. 
.88.-BEAMS  AN" 

their  Rf 


fOf 


89.— MODEF  rop- 

ertf  J.S.A. 

90 — ROr  ^ A-oscope. 

91.— LEVELING  :  Barometric,  Trigonometric  and  Spirit. 
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By  we 


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1.— CHIMNEYS  FOR  FURNACES,  FIRE- 
PLACES, AND  STEAM  BOILERS.  By 
R.  ARMSTRONG,  C.  E. 

2.— STEAM  BOILER  EXPLOSIONS.     By  ZE- 

RAH    COLBURN. 

3.— PRACTICAL  DESIGNING  OF  RETAIN- 
ING WALLS.     By  ARTHUR  JACOB.  A.  B, 
.  -PROPORTIONS     OF     PINS     USED     IN 
BRIDGES.     By  CHARLES  BENDER,  C.  E. 

5.— VENTILATION  OF  BUILDINGS.  By  W. 
F.  BUTLER. 

0.  -ON  THE  DESIGNING  AND  CONSTRUC- 
TION OF  STORAGE  RESERVOIRS. 
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7.— SURCHARGED  AND  DIFFERENT 
FORMS  OF  RETAINING  WALLS. 
By  JAMES  S.  TATE,  C.E. 
A  TREATISE  ON  THE  COMPOUND 
ENGINE.  By  JOHK  TUKNBULL.  Jr.  (In 
preparation.) 


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